Remark 3.1 Due to the cut off in (REF ), system (REF ) is not exact.
We called it an approximation model.
As mentioned in Remark REF , directly solving the inverse problem with the optimization method may be problematic since the stability is just conditionally logarithmic.
In contrast, by a step of approximation, we derive a system of elliptic equations with Cauchy data. It is well-known that solving elliptic equations with full boundary conditions is stable. This promises the success for our numerical study.
This cut off technique was used to solve several different types of inverse problems; see e.g. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}.
In contrast, proving the convergence of (REF ) as \(N \rightarrow \infty \)  is extremely challenging.
Studying the behavior of (REF ) as \(N \rightarrow \infty \)  is out of the scope of the paper.

